Experiment 8 ~ Rotational and Translational Energies Purpose: The objective of this experiment is to examine the conversion of gravitational potential energy to different types of energy: translational, rotational, and "internal" kinetic energy. Theory: Energy In: If an object of mass m slides or rolls down a total vertical distance Δh along an inclined plane, its gravitational energy decreases by Umg(Δh) during its downward trip. The lost gravitational potential energy is converted into other types of energy, such as kinetic energy. Note that if the object is traveling a distance L along the inclined plane, the total vertical distance it covers is ΔhL sin θ, where θ is the angle of the incline. Draw a sketch to convince yourself that this is true! Velocity Out: Given the time Δt to traverse a small interval Δd at the end of its trip, you can get a very rough estimate of the object s final velocity: vδd/δt. Kinetic Energy: An object of mass m will have a translational kinetic energy of (1) If the object is a ball or a cylinder, it will also have rotational kinetic energy!! Remember that (2) where r is the radius and I is the moment of inertia. The moment of inertia of an object depends on its shape and other properties, like whether it is solid or hollow. (See Table 9.1 in the textbook!) For a solid cylinder of radius r, the moment of inertia is (3) and so, if you substitute Eq. (3) into Eq. (2), you can show that the rotational kinetic energy will be. (4) (Do the calculations yourself to make sure you agree!) For a hollow cylinder of radius r, the moment of inertia is
(5) and substituting Eq. (5) into Eq. (2), the rotational kinetic energy becomes. (6) Again, check the algebra yourself to make sure you agree. Other Energy: You might think that, due to conservation of energy the initial potential energy should be converted completely into kinetic energy, which includes both rotational and translational energy. So, since you could write E initial E final, (7) UK t + K r. (8) But that s not quite true! Not all the initial potential energy is transformed into kinetic energy. Some of it is lost in other ways, to non-conservative forces like friction and drag (air resistance). So really, where ΔE is this extra energy. Equipment: UK t +K r +ΔE (9) The experimental setup is diagrammed in Figures 1 and 2, and consists of an elevated air track, photo gate timer, and a removable ramp for the rolling object measurements. You ll do measurements with three different objects: a glider (which will only have K t ), and two types of cylinder (solid and hollow). The cylinders will have both K t and K r. Use time in tab labeled time in gate. For the glider, the photo gate should be at the level of the flag. For the cylinders, the photo gate should be at a height as close as possible to the diameter of the cylinder (so that the distance that is crossing the photo gate is equal to the diameter of the cylinder).
Experiment: Glider: Determine the mass of the glider. Determine the total change in glider height Δh from its release at the top of the ramp to when it crosses the center point of the photo gate. The glider is going to travel a total distance L as shown in the figure below. You want to determine the height when it starts (point 1) and its height when it reaches the photo gate (point 2). These are the vertical heights shown by the red lines in the figure. The difference between these heights is Δh. Alternatively (if you like trigonometry), you can calculate Δh from the angle of the incline and the distance L: ΔhL sin θ. Measure the time interval for the glider to pass through the photo gate. Do this for 5 trials. Record the data in the table below. Solid cylinder: Determine the mass of the solid cylinder. Determine the total change in the cylinder s height Δh from its release at the top of the ramp to when it crosses the center point of the photo gate. Use the same procedure as for the glider to do this, measuring the height at the top of the ramp and the height at the bottom of the ramp, and taking their difference or you can calculate ΔhL sin θ. Measure the time interval for the solid cylinder to cross the photo gate. Do this for 5 trials. Record the data in the table below.
Hollow cylinder: Use the time between Gates for this part of the experiement. (From what you know about how the photo gate timers work, can you figure out why? Hint: Think about how the photo gate is triggered, and about what the hollow cylinder s cross section looks like compared to the cross section of the solid cylinder.) Determine the mass of the hollow cylinder. Determine the total change in the cylinder s height Δh from its release at the top of the ramp to when it crosses the center point of the photo gate. Use the same procedure as for the glider to do this, measuring the height at the top of the ramp and the height at the bottom of the ramp, and taking their difference or you can calculate ΔhL sin θ. (You might even want to check that both ways of calculating Δh give you the same number!) Measure the time interval for the hollow cylinder to cross the photo gate. Do this for 5 trials. Record the data in the table on the next page.
Analysis Part I: Glider: Length of object passing through photo gate (glider flag) d Δh (m) mass (kg) t (s) (m) Solid cylinder: v d/t avg Length of object passing through photo gate (cylinder diameter) d Δh (m) mass (kg) t (s) (m) v d/t avg Hollow cylinder: Length of object passing through photo gate (cylinder diameter) d Δh (m) mass (kg) t (s) (m) v d/t avg
Analysis Part II: Now you have the height and final velocity for each of the objects, and you can calculate the changes in their energies using the equations derived in the Theory section above. Glider: Potential energy Lost: U (formula) Translational Kinetic Energy Gained: K t (formula) % of initial potential energy Additional loss of energy due to friction: E U - K t % of initial potential energy Solid Cylinder: Potential energy Lost: U (formula) Translational Kinetic Energy Gained: K t (formula) % of initial potential energy Rotational Kinetic Energy Gained: K r (formula) % of initial potential energy Total Kinetic Energy: K tot K t + K r % of initial potential energy Additional loss of energy due to friction: E U - K tot % of initial energy
Hollow Cylinder: Potential energy Lost: U (formula) Translational Kinetic Energy Gained: K t (formula) % of initial energy Rotational Kinetic Energy Gained: K r (formula) % of initial energy Total Kinetic Energy: K tot K t + K r % of initial energy Additional loss of energy due to friction: E U - K tot % of initial energy Once you have completed your calculations you can check your work by downloading the Excel spread sheet from: http://www.umsl.edu/~physics/lab/assets/mechanics/rotationale.xls
Questions: 1. For the glider, are U and K t equal? Should they be equal? If there is a difference between them, what might be the cause of the difference? 2. For the solid cylinder, is there any mechanical energy lost to friction? 3. For the hollow cylinder, is there any mechanical energy lost to friction? 4. For which of the three objects is ΔE the largest? Can you give any hypothesis as to why? 5. Which object showed the most efficient conversion of gravitational potential energy to translational kinetic energy? Why? 6. Which object showed the most efficient conversion of gravitational potential energy to rotational kinetic energy? Why? 7. Which object showed the least efficient conversion of gravitational potential energy to total kinetic energy? Why? 8. List some sources of error in the experiment. What types of error are they? 9. Think about the way you calculated the final instantaneous velocity in this experiment. There is a way in which this measurement was cheating. Can you explain why? 10. Did you need to measure the mass of the objects in order to compare their energies? Explain why or why not? 11. For the glider part of the experiment, we used an air track. Why? Suppose you had the glider just sliding down the incline? What would have been different? How might that have affected your energy calculations?